arXiv subjects

Hung Le

Publications and source records attributed to Hung Le.

At least 19 recordsLinked to original sources

arXiv2026

Lotus: Learning to Understand Time-series by Learning to Forecast

Time-series language models typically align numerical signals with language using labels, captions, or question-answer pairs, without explicitly requiring their representations to preserve how the underlying signals evolve. We introduce forecast grounding, a pretraining principle that requires language-aligned time-series representations to remain predictive of the future evolution before they are adapted to downstream tasks. We instantiate this principle in Lotus, which divides each time-series channel into non-overlapping patches and maps them into the LLM embedding space. It then predicts the next patch from the LLM hidden states of the time-series tokens, encouraging the learned representation to retain temporal information. To capture both temporal changes and periodic patterns, we also propose a temporal-spectral patch representation that combines each raw time-series patch with its spectral representation before mapping it into the LLM embedding space. Across three language model backbones and six downstream evaluation datasets, Lotus achieves the best average performance with minimal trainable parameters, reaching 59.91% accuracy and outperforming the strongest matched baseline by 3.36%. Our representation analyses further show that Lotus uses the numerical input and preserves temporal structure in its learned patch representations. These results establish forecast grounding as a promising pretraining principle, enabling compact time-series language models to outperform models 25-50 times their size on TSRBench. Our code is publicly available at https://github.com/DA2I2-SLM/Lotus

cs.LG↗

arXiv2026

$K_h$-Minor-Free Graphs Have Balanced Separators of Size $O(h\sqrt{n})$

We show that every $K_h$-minor-free graph with $n$ vertices has a balanced separator of size $O(h\sqrt{n})$ for any parameter $h \in \mathbb{N}$, resolving the conjecture posed by Alon, Seymour, and Thomas [AST90] in 1990. The proof is elementary, building on the combinatorial framework by Le [Le26]. A key new idea of the proof is the existence of a short hitting walk on vertex expanders: given $k$ large (not necessarily disjoint) subsets of vertices, there is a walk of length $O(k/ψ(G))$ hitting every subset, where $ψ(G)$ is the vertex expansion. This technical result substantially improves on an analogous lemma by Alon, Seymour, and Thomas [AST90] in an important parameter setting, where the previous lower bound [Kot00] does not apply.

math.CO↗

arXiv2026

Efficient Best-Of-N with Radial Consensus Score

Solving complex problems with large language models (LLMs) often requires selecting the best answer from multiple candidate responses (best-of-$N$), yet identifying the most reliable one remains challenging, particularly when correctness does not align with majority agreement. Existing approaches based on voting or probability scoring rely on surface-level signals (e.g., frequency or token-level likelihood), often missing high-quality rare answers and ignoring the geometric structure of candidate representations, where correctness may emerge as semantic consistency rather than surface agreement. To address these limitations, we introduce Radial Consensus Score (RCS), a simple, efficient, and training-free method for best-of-$N$ selection. RCS models semantic consensus by computing a weighted Frechet mean (semantic center) of answer embeddings and ranking candidates by their radial distance to this center. This enables it to recover semantically coherent correct answers despite lexical fragmentation, even when no single surface form holds a plurality under exact-match voting. Importantly, RCS offers a flexible framework that supports multiple weighting schemes, enabling the integration of arbitrary signals, such as semantic agreement or model confidence, with the position of the center adapting accordingly. Extensive experiments across five open-weight models and various benchmarks demonstrate that RCS matches or outperforms strong baselines at every sampling budget, with the largest gains when correct answers are lexically fragmented. Moreover, RCS serves as an effective drop-in replacement for majority voting in multi-agent debate and remains effective in black-box scenarios. These results suggest that geometric consensus can serve as a scalable and reliable alternative to voting-based methods for answer selection. Code is publicly available at https://github.com/manhitv/RCS.

cs.CL↗

arXiv2026

Beyond Tokenization: Direct Timestep Embedding and Contrastive Alignment for Time-Series Question Answering

Recent advances in large language models (LLMs) have given rise to time-series question answering (TSQA), which formulates time-series analysis as natural-language question answering. However, directly feeding raw numerical series into LLMs suffers from a tokenization bottleneck: Byte Pair Encoding fragments continuous values into unstable tokens whose embeddings lack meaningful metric structure, resulting in the loss of magnitude, scale, and trend information. Prior methods use patch-based encoders that split the series into fixed windows, locking in one granularity that breaks patterns and hides exact timesteps, through a separate module that rarely transfers across datasets with different lengths or sampling rates. To address this challenge, we propose CADE (Contrastive Alignment with Direct Embedding), a novel framework for TSQA built upon two key components: direct timestep embedding and semantic alignment. The proposed framework maps each timestep directly into the LLM embedding space through a point-wise linear encoder and MLP projector, preserving exact index-level access while eliminating the need for patching and padding. To further bridge the semantic gap between time-series and language representations, we introduce a novel one-directional supervised contrastive loss that aligns time-series embeddings with frozen class-name text anchors. Experimental results on the public Time-MQA benchmark demonstrate that our framework consistently improves performance across six TSQA tasks, outperforming both open-source and proprietary LLM baselines.

cs.CL↗

arXiv2026

Think More or Read More? A Token-Budget Theory of Reasoning with Shared Context

Reasoning systems increasingly improve answers by incorporating additional context into the model's input, from multi-agent debate, where agents read their peers' responses, to self-refinement, where a model revisits its own drafts. In these systems, communication consumes context as one agent's output becomes another's input. We study the regime where token reading and generation draw on one fixed token window, so every token spent reading is a token that cannot be spent reasoning. We develop a \textbf{token-budget theory} that formalizes this trade-off, modeling accuracy as a function of the reasoning budget remaining after reading and aggregating votes with a Condorcet jury model that permits correlated errors. For multi-agent debate, the theory gives a closed-form prediction for the optimal number of agents. This quantity is estimated from single-agent measurements, allowing the optimal group size to be determined without running debate, and the same accounting extends to a heterogeneous set of different models. We empirically verify our theory and show that, at the predicted group size, our method outperforms strong baselines under the tightest calibration spend, where debate-calibrated rules can afford only one or two questions. Self-refinement obeys the same accounting, with its own draft in place of a peer message. The theory further predicts that communication pays only when the gain in the accuracy ceiling exceeds the reasoning room spent on reading, with this condition tightening as the window shrinks. Across five open models and four reasoning tasks, self-consistency matches or beats deployed debate at $93$\% of $980$ matched window-and-group-size settings, with the gap being largest under tight windows and nearly disappearing at a loose one, consistent with the theory. Under a shared window, how much room a model has left to \textit{think matters more than what it reads}.

cs.CL↗

arXiv2026

DREAM: Dynamic Resolution Assignment For Multimodal Multi-agent Debate

Multi-agent debate (MAD) has emerged as an effective paradigm to improve the reasoning capabilities of large language models (LLMs) and is increasingly being extended to multimodal settings. However, existing multimodal MAD frameworks typically expose agents to the same fixed visual input, ignoring substantial variation in the visual scale needed across samples and agents. In addition, these frameworks frequently suffer from groupthink, a phenomenon where agents prematurely abandon correct deductions to conform with confident but hallucinated peer responses. To address these bottlenecks, we introduce DREAM (Dynamic Resolution Assignment For Multimodal Multi-Agent Debate), which operates via two core components: (1) Dynamic Resolution Assignment, a zero-shot probe round where agents test multiple resolutions, quantify uncertainty using Average Normalized Log-Likelihood (ANLL), and use an adaptive threshold to assign each agent to its empirically optimal resolution; (2) Uncertainty-Guided Rollback Aggregation counters groupthink by tracking each agent's uncertainty over rounds and restoring early low-uncertainty answers overridden by group pressure. On six multimodal datasets, DREAM improves the accuracy-token trade-off over multi-agent debate baselines by 1.5-3.2% accuracy without dataset-specific tuning.

cs.AI↗

arXiv2026

Metric Planarity Testing is Hard

A key milestone in algorithmic graph theory is Hopcroft and Tarjan's 1974 linear-time algorithm for graph planarity testing. At a workshop in 2002, Linial and Rabinovich asked for either a polynomial-time algorithm or an NP-hardness proof for metric planarity testing: given a metric space $(X,d_X)$, determine whether there exists an edge-weighted planar graph $G=(V,E,w)$ with $X\subseteq V$ that realizes the distances in $X$, meaning that $d_G(x,y)=d_X(x,y)$ for every $x,y\in X$. This question arose in the context of research on metric embeddings and their applications in computer science since the mid-1990s. Over the past few decades, polynomial-time algorithms have been developed for several special cases. In this work, we show that metric planarity testing is NP-hard, in sharp contrast to graph planarity testing. Our reduction yields a stronger result: under the Exponential Time Hypothesis (ETH), no $2^{o(n)}$-time algorithm for metric planarity testing exists, where $n$ is the number of points in the metric. To complement our ETH lower bound, we give two algorithmic results: (i) an algorithm for computing a metric planar embedding in $2^{\tilde{O}(n^2)}$ time and (ii) a $(1+ε)$-approximation algorithm in $2^{\tilde{O}(n)}$ time for any fixed $ε\in(0,1)$. The second algorithm produces a planar graph that preserves the input metric up to a factor of $(1+ε)$.

cs.DS↗

arXiv2026

Uncertainty-Aware Budget Allocation for Adaptive Test-Time Reasoning

Sampling multiple responses improves language model reasoning, but uniform compute allocation is inefficient because easy questions are over-sampled while hard questions remain under-explored. We propose \textbf{Uncertainty-Aware Budget Allocation (UAB)}, a concave integer optimization framework that reallocates a fixed sampling budget using uncertainty estimated from the initial samples themselves. In Phase-1, every question receives a small fixed number of generations. Their answer disagreement, measured by vote entropy, provides a difficulty signal while these generations contribute to the final vote. In Phase-2, the remaining budget is allocated by a marginal-greedy algorithm that optimally solves a concave coverage-maximization surrogate, concentrating samples on questions whose initial answers disagree. Across five open-weight models (1.5B--27B parameters) and five reasoning benchmarks of varying difficulty, UAB improves average accuracy by $+2.3\%$ over uniform allocation, and achieves gains of up to $+5.5\%$ on individual benchmarks, with the largest gains in low-resource settings. Moreover, UAB meets the target budget exactly and requires no auxiliary model or additional LLM calls. Code is publicly available at https://github.com/manhitv/UAB.

cs.CL↗

arXiv2026

SteerCast: Retrieval-Based Latent Steering for Decoder-Only Time Series Forecasting

Time series forecasting aims to predict future values from historical observations and auxiliary features. We propose \textbf{SteerCast}, a retrieval-based latent steering method that improves decoder-only forecaster at inference time, without updating its parameters. SteerCast constructs a database from the training set by storing a representation of each history window together with a \emph{steering vector} computed in the forecaster's latent space, defined as the difference between representations induced by the ground-truth continuation and by the model's own prediction. At test time, SteerCast retrieves nearest neighbors for a query history, aggregates their steering vectors, and injects the resulting signal into the forecaster's hidden states at every step of autoregressive generation, guiding predictions toward trajectories consistent with similar training cases. Experiments across diverse multivariate benchmarks and multiple horizons show that SteerCast consistently improves forecasting accuracy over the fine-tuned backbone and retrieval-based baselines, while requiring no additional training beyond the original fine-tuning and using only the training set as a retrieval corpus.

cs.LG↗

arXiv2026

Peeling Half the Onion: Embedding $k$-Outerplanar Graphs into $\ell_1$ and Trees with a Polynomial Distortion (in $k$)

Chekuri, Gupta, Newman, Rabinovich, and Sinclair [SODA'03] showed that $k$-outerplanar graphs can be embedded into trees and $\ell_1$ with distortion $2^{O(k)}$. Their result is perhaps the strongest evidence supporting the still-open planar $\ell_1$-embedding conjecture: planar metrics can be embedded into $\ell_1$ with constant distortion. However, the exponential dependency on $k$ in their distortion bound remains the state of the art. Their embedding is obtained by a so-called onion-peeling approach: removing one layer of the graph at a time at the cost of incurring $O(1)$ distortion multiplicatively, and recursively embedding the resulting $(k-1)$-outerplanar graph. In this paper, we devise a new onion peeling approach that peels $k/2$ layers off the input $k$-outerplanar graph at every step. As a result, we obtain the first embedding of $k$-outerplanar graphs into trees and $\ell_1$ with distortion $\operatorname{poly}(k)$, an exponential improvement over the best-known distortion bound [SODA'03]. Our result of embeddings into trees comes closer to the distortion lower bound $Ω(k)$ for $k$-outerplanar graphs.

cs.DS↗

arXiv2026

Volatility-Clustering Adaptation for Financial Time Series

Time-series foundation models are increasingly adapted to new domains through fine-tuning on target data, under the implicit assumption that more target data yields better forecasts. We show that this assumption can fail in financial forecasting, where individual price changes are difficult to predict, but large moves tend to cluster, creating alternating calm and turbulent periods. Using financial foundation models trained on price bars of open, high, low, close, and volume, we argue that adapting to financial domains requires training signals beyond next-token prediction. We introduce Volatility-Clustering Adaptation (VCA), which augments next-token cross-entropy with a differentiable penalty on the autocorrelation of squared returns, the standard statistical signature of volatility clustering. This additional objective provides a multi-step training signal by matching the resulting dependence structure of autoregressive rollouts to those of the realized future. Across three asset sets and two evaluation conventions, VCA improves adaptation over the pre-trained model, with the strongest gains under the primary evaluation (\textsc{fore}), driven primarily by reduced variance error. Overall, our results suggest that effective financial adaptation requires objectives that capture domain-specific temporal structure beyond token-level prediction.

cs.LG↗

arXiv2026

Separator Theorem for Minor-Free Graphs in Linear Time

The planar separator theorem by Lipton and Tarjan [FOCS '77, SIAM Journal on Applied Mathematics '79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan's theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC '90, Journal of the AMS '90] showed that any minor-free graph admits a balanced separator of size $O(\sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(\sqrt{n})$ or have superlinear running time, or both. In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.

cs.DS↗

arXiv2026

Retrieval-augmented Decoding for Improving Truthfulness in Open-ended Generation

Ensuring truthfulness in large language models (LLMs) remains a critical challenge for reliable text generation. While supervised fine-tuning and reinforcement learning with human feedback have shown promise, they require a substantial amount of annotated data and computational resources, limiting scalability. In contrast, decoding-time interventions offer lightweight alternatives without model retraining. However, existing decoding strategies often face issues like prompt sensitivity, limited generalization, or dependence on internal model states. We propose Retrieval-Augmented Decoding (RAD), a context-aware adaptive decoding method that leverages a compact reference grounding space built from as few as 10 annotated examples and comprising pairs of context embeddings and next-token logits from truthful responses, to enable retrieval-based logit shaping during inference. At each decoding step, RAD retrieves high-quality semantically similar contexts from this grounding space and aggregates their associated next token logits to modify the model's current logits. Across four open-ended generation benchmarks and four LLMs, our method consistently outperforms strong baselines and shows robust cross-task generalization, underscoring the promise of context-aware decoding for enhancing factual reliability.

cs.LG↗

arXiv2026

Memory in Deep Time-Series Models

Deep learning for time series has progressed through successive architectural paradigms, from recurrent networks and transformers to structured state-space models, retrieval-augmented predictors, foundation models, and tool-using agents. These developments are typically studied in isolation, organized by architecture or modeling era. We argue that they can instead be viewed through a common question of \emph{how does a time-series model retain and access information beyond its immediate input?} This question is motivated by a fundamental limitation of conventional time-series modeling: information relevant to a prediction may lie far beyond a feasible input window, while compressing history into a fixed-size state can discard information that may become useful later. We formulate this challenge as a \emph{memory} problem and organize existing time-series methods along a spectrum from internal memory, encoded in parameters and fixed-size states, to external memory that is addressable, retrievable, and increasingly maintained by agents. We then develop a unified taxonomy of memory mechanisms and review three classes of external memory, including explicit modules, retrieval augmentation, and agentic stores, under a common framework for what is retained, how it is written and accessed, and how it persists. A cross-cutting analysis maps these mechanisms to time-series tasks and identifies gaps in both methods and evaluation. We conclude by outlining open problems in building memory systems that can selectively retain, retrieve, revise, and forget information as temporal environments evolve. The result is a framework for studying memory as a first-class dimension of time series modeling, independent of the underlying backbone.

cs.LG↗

arXiv2026

Creative Generation via Multi-Agent Debate: Does Debate Suppress Diversity?

Creative generation tasks, such as narrative writing and scientific ideation, demand both high-quality outputs and distinct responses across independent runs to maximize exploration. Multi-Agent Debate (MAD) has shown strong quality gains on factual and reasoning tasks, making it a natural candidate for creative generation. However, we find its convergence-driven design actively suppresses output diversity across independent runs, creating an inherent trade-off with creative tasks. We theoretically show that preserving diversity among agents within each debate session is a necessary condition for achieving diverse outputs across independent runs. Building on this finding, we propose Creative-MAD, which introduces two synergistic interventions to sustain agent divergence. Specifically, Cognitive Lens Assignment counters identity drift by anchoring each agent to a distinct and persistent cognitive mode, while Embedding-based Peer Selection counters majority pull by limiting each agent's context to its most semantically distant peers. Experiments across four creative benchmarks demonstrate that Creative-MAD significantly enhances both lexical and semantic diversity while maintaining MAD's output quality.

cs.CL↗

arXiv2026

Three trees suffice for a constant stretch in minor-free graphs

In this short note, we show that $H$-minor-free graphs have a tree cover with $3$ trees and constant stretch for any fixed graph $H$. The number of trees matches the recent lower bound by Chen, Tan, and Xu who showed that a toroidal grid requires at least $3$ trees for constant stretch. Our result is obtained by establishing a connection between tree covers and Assouad--Nagata dimension and then invoking the recent dimension bound for minor-free metrics by Liu.

cs.DS↗

arXiv2026

Improved Euclidean Shallow Light Trees

For parameters $α,β\geq 1$, a spanning tree $T$ of a weighted graph $G$ rooted at a designated vertex $r$ is called an $(α,β)$-shallow-light tree (SLT) if (i) for every vertex $v$, $d_T(r,v) \leq α\cdot d_G(r,v)$ (root-stretch $α$), and (ii) $w(T) \leq β\cdot w(\mathsf{MST})$ (lightness $β$). The pioneering work of Khuller, Raghavachari, and Young (SODA 1993) constructed $\left(1+ε, \tfrac{2}ε+1\right)$-SLTs for general weighted graphs, and proved that this tradeoff between root-stretch and lightness is tight even for series-parallel graphs. They further asked whether even a slight improvement, namely reducing the lightness to $\tfrac{2-c}ε$ for any constant $c>0$, is possible in the Euclidean plane. We resolve this longstanding question in the affirmative. Specifically, we show that every Euclidean instance admits an SLT with root-stretch $1+ε$ and lightness at most $\left(\frac{5}{3} + o_ε(1)\right) \cdot \frac{1}ε$, thereby significantly improving upon the longstanding $2/ε$ barrier. As our second main result, we provide a construction of SLTs in the Euclidean plane, with root stretch $1+ε$ and lightness at most $\left(\frac{2π}{\sqrt{4π^2+1}}+o_ε(1)\right)\frac{1}ε \approx (0.987+o_ε(1))\frac{1}ε$. Notably, this reduces the leading $2/ε$ term in the lightness bound by more than a factor of two, and comes quite close to the lower bound of $\left(\frac{2π}{2π+1} +o_ε(1))\right) \cdot \frac{1}ε \approx (0.862 +o_ε(1))\frac{1}ε$ by Elkin and Solomon (FOCS 2011).

cs.CG↗

arXiv2026

LLM as Forecasting Planner: Training-Free Text Conditioning for Time-Series Foundation Models

Text-conditioned time-series forecasting predicts a series from both its numerical history and natural-language context, allowing forecasts to account for events and constraints that the past alone cannot reveal. This requires both reliable numerical forecasting and the ability to interpret contextual information. Time-series foundation models (TSFMs) provide strong numerical forecasts, while large language models (LLMs) can reason over text, but combining their strengths remains challenging because asking an LLM to generate or revise forecast values directly can distort the temporal structure captured by the TSFM. We instead formulate forecasting as a planning problem over TSFM-generated trajectories. The frozen TSFM acts as a simulator that proposes numerical continuations, while the LLM acts as a policy and value function that guides candidate selection and evaluates completed trajectories against the context. We instantiate this as \rc{} (\textbf{L}LM \textbf{A}s \textbf{F}orecasting \textbf{P}lanner), a training-free framework that bridges the modality gap without retraining either model, using Monte Carlo tree search (MCTS) over the forecast horizon with a \emph{Ranker} LLM as policy and a \emph{Judge} LLM as value function. Experiments on Context-is-Key and Time-MMD across two TSFM backbones (Chronos and TimesFM) and four LLMs show that \rc{} delivers consistent improvements across model choices, supporting sequential search as an effective training-free approach to text-conditioned forecasting.

cs.LG↗